Juicing beets?!?

Without a juicer, this is probably your best bet:
1) remove stalk, peel outside skin... Prepare for stained hands or get gloves so it doesn't look like you murderer someone/ have jaundice (depending on red or yellow beets)
2) use a food processor to get as close to a purée as you can
3) separate pulp and juice--best cheap method is using a nut milk bag or cheese cloth. They sell them at whole foods and the like for a couple of bucks. If you have a fine sieve, that will work. A chinois would also work really well if you happen to have one of those.

If you're really desperate and don't have a food processor or any sort of sieve you can get really creative. Microplane and paper towels. Microplane is your food processor and paper towels are your sieve. Be prepared for a little extra mess though. I've done it before though when visiting other people with ginger.

Are you after raw juice for drinking? If not you can chop it Very finely and heat it over very low heat with just a tad of water until it releases it's own juice. Cool and strain.