Unless you have a time machine, you won't be able to watch all 798 episodes of Doctor Who in time to catch up to today's special 75-minute episode commemorating the show's 50th anniversary. In part you'd need a time machine because nearly 80 episodes were destroyed in the BBC archives (though some of them still survive as audio clips thanks to fans who recorded the transmissions) but mainly because you wouldn't have enough time.
In order to catch up on longest running sci-fi television show in history (take that, Star Trek), you would need to take about month of sick days, find a bowl of popcorn the size of an Olympic swimming pool, and maybe invest in a pair of glasses, because you're going to strain your eyes something serious. With half a century of Doctors, companions, extraterrestrial characters, time traveling escapades, and millions of worlds to see, you've got a long way to go. Or you could take the short cut (spoilers) and jump into Everything You Need to Know About Doctor Who. Either way, let's do this. Hop in my TARDIS, shorty.