Yes it is possible, just redirect the output (AKA stdout) to a file:

SomeCommand > SomeFile.txt  

Or if you want to append data:

SomeCommand >> SomeFile.txt

If you want stderr as well use this:

SomeCommand &> SomeFile.txt  

or this to append:

SomeCommand &>> SomeFile.txt  

if you want to have both stderr and output displayed on the console and in a file use this:

SomeCommand 2>&1 | tee SomeFile.txt

(If you want the output only, drop the 2 above)

To write the output of a command to a file, there are basically 10 commonly used ways.

Overview:

_{Please note that the n.e. in the syntax column means "not existing".
There is a way, but it's too complicated to fit into the column. You can find a helpful link in the List section about it.}

          || visible in terminal ||   visible in file   || existing
  Syntax  ||  StdOut  |  StdErr  ||  StdOut  |  StdErr  ||   file   
==========++==========+==========++==========+==========++===========
    >     ||    no    |   yes    ||   yes    |    no    || overwrite
    >>    ||    no    |   yes    ||   yes    |    no    ||  append
          ||          |          ||          |          ||
   2>     ||   yes    |    no    ||    no    |   yes    || overwrite
   2>>    ||   yes    |    no    ||    no    |   yes    ||  append
          ||          |          ||          |          ||
   &>     ||    no    |    no    ||   yes    |   yes    || overwrite
   &>>    ||    no    |    no    ||   yes    |   yes    ||  append
          ||          |          ||          |          ||
 | tee    ||   yes    |   yes    ||   yes    |    no    || overwrite
 | tee -a ||   yes    |   yes    ||   yes    |    no    ||  append
          ||          |          ||          |          ||
 n.e. (*) ||   yes    |   yes    ||    no    |   yes    || overwrite
 n.e. (*) ||   yes    |   yes    ||    no    |   yes    ||  append
          ||          |          ||          |          ||
|& tee    ||   yes    |   yes    ||   yes    |   yes    || overwrite
|& tee -a ||   yes    |   yes    ||   yes    |   yes    ||  append

List:

• command > output.txt

  The standard output stream will be redirected to the file only, it will not be visible in the terminal. If the file already exists, it gets overwritten.

• command >> output.txt

  The standard output stream will be redirected to the file only, it will not be visible in the terminal. If the file already exists, the new data will get appended to the end of the file.

• command 2> output.txt

  The standard error stream will be redirected to the file only, it will not be visible in the terminal. If the file already exists, it gets overwritten.

• command 2>> output.txt

  The standard error stream will be redirected to the file only, it will not be visible in the terminal. If the file already exists, the new data will get appended to the end of the file.

• command &> output.txt

  Both the standard output and standard error stream will be redirected to the file only, nothing will be visible in the terminal. If the file already exists, it gets overwritten.

• command &>> output.txt

  Both the standard output and standard error stream will be redirected to the file only, nothing will be visible in the terminal. If the file already exists, the new data will get appended to the end of the file..

• command | tee output.txt

  The standard output stream will be copied to the file, it will still be visible in the terminal. If the file already exists, it gets overwritten.

• command | tee -a output.txt

  The standard output stream will be copied to the file, it will still be visible in the terminal. If the file already exists, the new data will get appended to the end of the file.

• (*)

  Bash has no shorthand syntax that allows piping only StdErr to a second command, which would be needed here in combination with tee again to complete the table. If you really need something like that, please look at "How to pipe stderr, and not stdout?" on Stack Overflow for some ways how this can be done e.g. by swapping streams or using process substitution.

• command |& tee output.txt

  Both the standard output and standard error streams will be copied to the file while still being visible in the terminal. If the file already exists, it gets overwritten.

• command |& tee -a output.txt

  Both the standard output and standard error streams will be copied to the file while still being visible in the terminal. If the file already exists, the new data will get appended to the end of the file.

You can also use tee to send the output to a file:

command | tee ~/outputfile.txt

A slight modification will catch stderr as well:

command 2>&1 | tee ~/outputfile.txt

or slightly shorter and less complicated:

command |& tee ~/outputfile.txt

tee is useful if you want to be able to capture command output while also viewing it live.

You can redirect the command output to a file:

your_command >/path/to/file

To append the command output to a file instead of overwriting it, use:

your_command >>/path/to/file

An enhancement to consider -

Various scripts will inject color codes into the output which you may not want cluttering up your log file.

To fix this, you can use the program sed to strip out those codes. Example:

command 2>&1 | sed -r 's/'$(echo -e "\033")'\[[0-9]{1,2}(;([0-9]{1,2})?)?[mK]//g' | tee ~/outputfile.txt

The script command

There are two different questions here. The first is in the title:

How do I save terminal output to a file?

The second question is in the body:

How do I save the output of a command to a file?

All the answers posted here address the second question but none address the first question which has a great answer in Unix & Linux:

• Save all the terminal output to a file

This answer uses a little known command called script which saves all your shell's output to a text file until you type exit. The command output still appears on your screen but also appears in the text file.

The process is simple. Use:

$ script ~/outputfile.txt
Script started, file is /home/rick/outputfile.txt
$ command1
$ command2
$ command3
$ exit
exit
Script done, file is /home/rick/outputfile.txt

Then look at your recorded output of commands 1, 2 & 3 with:

cat ~/outputfile.txt

This is similar to earlier answer of:

command |& tee ~/outputfile.txt

• But you don't have to use |& tee ~/outputfile.txt after each commnd.
• The script command has added benefit (or disadvantage) of reloading ~/.bashrc when it starts.
• The script command shows the command prompt ($PS1) followed by the command(s) you entered.
• The script command records all the details in full color.

Send output to clipboard

Many times we want the output to go to the clipboard so we can paste it later. From this answer you can use:

cat ~/.bashrc | xclip -selection clipboard

Now you can use Ctrl+V in almost any application to paste the terminal output into your document. To paste the terminal output in the clipboard back into your terminal use Ctrl+Shift+V instead.

some_command | tee command.log and some_command > command.log have the issue that they do not save the command output to the command.log file in real-time.

To avoid that issue and save the command output in real-time, you may append unbuffer, which comes with the expect package.

Example:

sudo apt-get install expect
unbuffer some_command | tee command.log
unbuffer some_command > command.log

Assuming log.py contains:

import time
print('testing')
time.sleep(100) # sleeping for 100 seconds

you can run unbuffer python log.py | tee command.log or unbuffer python log.py > command.log

More information: How can I save a command output to a file in real-time?

For cron jobs etc you want to avoid the Bash extensions. The equivalent POSIX sh redirection operators are

Bash          POSIX
------------  --------------
foo &> bar    foo >bar 2>&1
foo &>> bar   foo >>bar 2>&1
foo |& bar    foo 2>&1 | bar

You'll notice that the POSIX facility is in some sense simpler and more straightforward. The &> syntax was borrowed from csh which should already convince you that it's a bad idea.

If you want to output to the file while the command is being run:

script -c ./path/to/executable.bash -f log.txt

Use terminal emulator features

An option not mentioned yet, that can save colours / colors too, is to use a console program — such as Konsole (KDE/Plasma's default terminal emulator) — to save the output.

Konsole

Konsole has: 
File > Save output as... 

the shortcut is Ctrl+Shift+S; it allows the output to be saved as a text file, or as HTML including colors! I'm not sure exactly how much it will save but in my test it only included ~1000, the entire terminal scrollback, (you can increase the buffer by creating a new profile, Profile > New..., and then change the Scrolling settings to capture more; Konsole version 4:21.08.1).

gnome-terminal

gnome-terminal has "copy output as HTML" too, which allows pasting the HTML into a document; it preserves colour but only copies the content of the output currently shown on screen AFAICT.

generically

You can, of course, do a straight drag-select (hold left mouse button whilst dragging) and then copy (ctrl+c, Edit > Copy, or right-mouse-click and choose copy).

others?

Feel free to modify this answer to include other popular terminal apps. My favourite, Yakuake, does not appear to have this feature nor did most of the popular terminals I reviewed, including terminal.app and Hyper.