Yes, you can purchase digital downloads. The only place to purchase digital downloads of the Beatles catalog is from the Apple iTunes Store.
Most of the Beatles catalog is available for sale in the Apple iTunes store, in AAC format at 256 kbps. The Beatles tracks, like all other music sold on iTunes, is DRM-free. You can use the iTunes app for Mac or Windows to convert iTunes files into MP3 format, or to transcode the files into several other formats including burning them to CD.
There is a very long, strange story behind this: the Beatles, and specifically George Harrison (1943-2001), from 1978 onward, spent decades trying to sue Apple, Inc. out of business, because the Beatles' corporation, which they formed in 1968, was called Apple Corps, and the Beatles' record label, affiliated with Apple Corps, was called Apple Records. Apple Computer, which of course had nothing to do with Apple Records, was founded in the USA in 1976. George Harrison was incensed that there was a brand of computer hardware called "Apple" that could be used by musicians to record and edit music, or that could be used by consumers to play back music. Harrison, to his death, insisted that any use of the term "Apple" involving any kind of physical product that could make or play music, anywhere in the world, was an infringement on the trademarks of the Apple Records company which he co-founded.
The Beatles' Apple Records had lawsuits against Apple Computer tied up in British courts for decades, claiming that Apple Computer (later called Apple, Inc.) should abandon their name and trademarks. It proved impossible for Apple to completely settle all the lawsuits while George Harrison was still alive.
When Apple Computer opened the Apple iTunes Store in 2003 to sell digital downloads of music (even though none of the Beatles' music was sold there), the Beatles redoubled their legal efforts to sue Apple, saying that an online music store run by another company called "Apple" was unacceptable to them.
In fact it took another six years after Harrison's death before Apple could settle with Paul McCartney, Ringo Starr and the other rights-holders associated with the British Apple Records corporation.
In 2007 Apple, Inc. settled the Beatles' lawsuits by paying the Beatles' corporation a sum reported to be around US $500 million (earlier settlements for earlier disputes had already cost Apple, Inc. hundreds of millions of dollars as well.) The settlement included an arrangement for a licensing agreement of the necessary trademarks. Three years later, in 2010, the Beatles' catalog of recordings were made available for sale on iTunes.
As a result of this massive settlement between the two parties, so far digital downloads of the Beatles' catalog are only sold through the iTunes store, and nowhere else in the world.
Here is the Wikipedia article on Apple Corps v Apple Computer.