How do I remotely control the email client on my work Macintosh
(either  Eudora, Entourage, or Apple
Mail) from my PowerBook at home?

Try VNC.  There's been a port to OS X.  It basically lets you access
your computer from off-site as if you were directly on it (like remote
desktop).  It's free.

Do a Google search for it.

I have tried VNC. It is too slow to be useful, even over a broadband
connection. I also found on Google a link to a program that might do
the trick (Systeris from Rampbell Software), but the program appears
to have been discontinued. As advertised, this program allows you to
access your e-mail via from a Web site. What you are actually doing is
remotely manipulating your Entourage client. This might be faster than
VNC, which allows you to manipulate your entire computer remotely,
overkill in my case. Unfortunately, the Systeris program appears to
have died in 2002 when it was in the 0.9 beta version.

You can: sync to a PDA. Sync to an online mailbox. Go to
www.mail2web.com to check for new mail. Go to your ISP's webmail
account.

While it doesn't quite qualify as "controlling" your email, you might
be able to access your mail server remotely. If your mail server (POP3
or IMAP) is exposed to the internet (or via VPN) then you can
configure an email client at home to point to that server. If it is an
IMAP server then all of your mail is stored on the server and it
should be pretty straight forward. If you use a POP3 server then you
might want to configure your email clients at work and at home to
leave the mail on the server for a week or so. That way you can read
new emails from home or work. This might not be exactly what you were
hoping to accomplish but it might get you closer to your goal.

Use a yahoo email account which you can set up to check POP and IMAP
mailboxes.  I presume this is what your home mail client is checking? 
Be sure to set each email client you use to check the email to leave
the clients on the server if going this route, as suggested earlier.

IMAP (or Exchange) is in principle a solution to my problem. However,
my institution's IMAP server limits my storage to 250 MB. I would need
at least 2 GB for IMAP to be an option. I can get a 2GB account at
Yahoo! but I have to pay a fee for it. I could also convert my Mac OSX
system into my private IMAP server, but my institution's policies
expressly forbid setting up individual mail servers. The only  thing
that would work is: 1. Downloading all my email to my work computer
(where I have unlimited space) using any standard e-mail client such
as Eudora, Entourage, or Apple Mail, combined with 2. software that
allows me to interact with my e-mail program at work so that I can
view mesagges and reply to them remotely, but all activity and mailbox
changes are recorded on the work computer. So when I return to my
office on Monday, it looks as if I had checked and replied to my
e-mails while seating in front of my work computer, even though I
spent all weekend replying to e-mails from my home PowerBook.

I'm just curious.. are you seriously working on 2GB of data at any
given time? You can't IMAP that successfully anyway... I mean, you
can, but each time you load those attachments ... whew! Set yourself
an FTP server for the attachments and use the mail for mail.