impossible...
That's just not possible for me - from the moment I turn my computer on in the morning, to when I go to bed at night, I'm inescapably linked to Google.
I use GMail to check all of my email accounts, and I have Google Talk running all the time. I use the incredibly lovely Google Desktop sidebar to display my RSS feeds, calendars, and latest emails.
While at work, I check my email using the extremely useful GMail java app for mobile phones, and occasionally, I use the Google Maps app too (I'm a bicycle courier).
To go 24 hours without Google?
/Maybe/ if I only don't use it for searching for anything...