How to redirect output of an entire shell script within the script itself?

Is it possible to redirect all of the output of a Bourne shell script to somewhere, but with shell commands inside the script itself?

Redirecting the output of a single command is easy, but I want something more like this:

#!/bin/sh
if [ ! -t 0 ]; then
    # redirect all of my output to a file here
fi

# rest of script...

Meaning: if the script is run non-interactively (for example, cron), save off the output of everything to a file. If run interactively from a shell, let the output go to stdout as usual.

I want to do this for a script normally run by the FreeBSD periodic utility. It's part of the daily run, which I don't normally care to see every day in email, so I don't have it sent. However, if something inside this one particular script fails, that's important to me and I'd like to be able to capture and email the output of this one part of the daily jobs.

Update: Joshua's answer is spot-on, but I also wanted to save and restore stdout and stderr around the entire script, which is done like this:

# save stdout and stderr to file 
# descriptors 3 and 4, 
# then redirect them to "foo"
exec 3>&1 4>&2 >foo 2>&1

# ...

# restore stdout and stderr
exec 1>&3 2>&4

Addressing the question as updated.

#...part of script without redirection...

{
    #...part of script with redirection...
} > file1 2>file2 # ...and others as appropriate...

#...residue of script without redirection...

The braces '{ ... }' provide a unit of I/O redirection. The braces must appear where a command could appear - simplistically, at the start of a line or after a semi-colon. (Yes, that can be made more precise; if you want to quibble, let me know.)

You are right that you can preserve the original stdout and stderr with the redirections you showed, but it is usually simpler for the people who have to maintain the script later to understand what's going on if you scope the redirected code as shown above.

The relevant sections of the Bash manual are Grouping Commands and I/O Redirection. The relevant sections of the POSIX shell specification are Compound Commands and I/O Redirection. Bash has some extra notations, but is otherwise similar to the POSIX shell specification.

Typically we would place one of these at or near the top of the script. Scripts that parse their command lines would do the redirection after parsing.

Send stdout to a file

exec > file

with stderr

exec > file                                                                      
exec 2>&1

append both stdout and stderr to file

exec >> file
exec 2>&1

As Jonathan Leffler mentioned in his comment:

exec has two separate jobs. The first one is to replace the currently executing shell (script) with a new program. The other is changing the I/O redirections in the current shell. This is distinguished by having no argument to exec.

You can make the whole script a function like this:

main_function() {
  do_things_here
}

then at the end of the script have this:

if [ -z $TERM ]; then
  # if not run via terminal, log everything into a log file
  main_function 2>&1 >> /var/log/my_uber_script.log
else
  # run via terminal, only output to screen
  main_function
fi

Alternatively, you may log everything into logfile each run and still output it to stdout by simply doing:

# log everything, but also output to stdout
main_function 2>&1 | tee -a /var/log/my_uber_script.log

For saving the original stdout and stderr you can use:

exec [fd number]<&1 
exec [fd number]<&2

For example, the following code will print "walla1" and "walla2" to the log file (a.txt), "walla3" to stdout, "walla4" to stderr.

#!/bin/bash

exec 5<&1
exec 6<&2
 
exec 1> ~/a.txt 2>&1

echo "walla1"
echo "walla2" >&2
echo "walla3" >&5
echo "walla4" >&6

[ -t <&0 ] || exec >> test.log