Since I could not find an existing, small program, I decided to write my own one. Originally, I came up with:

#!/bin/bash
# Simple sendmail
# filename per spec at http://cr.yp.to/proto/maildir.html
rand=$((RANDOM % 1000))
msgname=$(date +%s).P$$R$rand.$(hostname | tr '/:' '\057\072')

# Safety measure: do not overwrite existing mail
set -o noclobber

cat > ~peter/".local/share/local-mail/inbox/new/$msgname"

That worked... except if the user executing the script is not "peter". The successor of this idea is a "small" C program, femtomail. From its README:

femtomail - minimal MDA with Maildir support

femtomail is a minimal Mail Delivery Agent (MDA) for local mail. Mail is accepted from standard input and placed in a Maildir box of a user. This software is intended for use on a single-user machine.

Remote delivery, daemonizing, sender verification, etc. is not implemented and won't be implemented due to its complexity. femtomail is not written because mail software did not exist, but because existing software were too large for the simple task of delivering cron mail to the local user.

The workflow of femtomail:

1. Change the process user and group.
2. Create a new file with a unique filename in the mail directory.
3. Write a Received header to the file.
4. Pass data from standard input to the file.
5. Exit.

The source code and installation instructions are available at https://git.lekensteyn.nl/femtomail/.

Cron will attempt to send an email with any output that may have occurred when the command was run. From cron's man page:

When executing commands, any output is mailed to the owner of the crontab (or to the user specified in the MAILTO environment variable in the crontab, if such exists). Any job output can also be sent to syslog by using the -s option.

So to disable it for a specific crontab entry just capture all of the commands output and either direct it to a file or to /dev/null.

30 * * * * notBraiamsBackup.sh >/dev/null 2>&1