Can I send money from the U.S. to Russia using PayPal or a similar service?

Of course you can. If opting for PayPal, then needless to say, the receiving party also needs PayPal. If opting for larger amounts, your best bet would be a Bank wire transfer (via SWIFT). You can also use money transfer services like Western Union, RIA, MoneyGram, etc. (but they will be slightly more expensive, but the money would be available almost instantaneously).