Days worth. Maybe weeks. Actually, probably months or years.
A bit over 15 days if you don't take any breaks. You can cut it down a bit by skipping over credits, the repeats of last week's cliffhanger in many episodes, etc. Of course you will die from thirst, from bladder problems, or at your wife's or mother's hands long before that 15 days is up.
Realistically, if you're only watching a few hours/night, taking reasonable breaks during watching, not watching every single night, etc., you're talking a good part of a year.
I know someone who actually watched ever classic story with no missing episodes over the course of 8 months, so maybe another month or two for the reconstructions, TV movie, and new series. At that point, it's hard to imagine you're not going to watch the Torchwood and Sarah Jane, the BBV and Reeltime spinoff videos, etc. So let's call it a year.
But that's assuming the people in white coats don't interrupt by carting you off to the funny-farm!
I began watching all of the episodes in order, since 2009. It is now 2012, and I only just finished watching that bloody long-winded story, The Daleks. Oh well, at least I've got the nice, short, Edge of Destruction to watch now. But I still have to get through Marco Polo, ahhhhhhh!
I'm currently nearing the end of my goal to watch all Doctor Who episodes (last half of Series 6, and 7), and at four Classic episodes, and then New episodes, it took me several months to get through the entire series.
I started watching Doctor who just under 2 years ago when I watched all revived episodes in about 2 weeks, then watched all confidentials, then all Torchwood, then all SJA, and then started watching the classic show starting from season 7 as there were no more missing episodes after that point. I've been watching those classic episodes on and off over the past year and a half (maybe longer) and have only just reached "The Five Doctors" as of a couple of weeks ago. So, yeah, it takes a while, but it really depends on how often you're watching. If you were dedicated and were watching all day everyday, you could finish a typical season in 1 day, so you could conceivably watch all classic episodes in 26 days and then 6 1/2 days for the new series after that. But only if you're watching for basically all day everyday, which probably isn't the best way to do it. As good as Doctor Who is, I doubt you will want to watch ~13 hours of episodes everyday, and it makes it quite hard to take in each story individually. The plot line kind of fall into one big blur if you try and think of the plot line of an individual episode. It's much more enjoyable to spread them out, maybe watch only a story or 2 per day. Anyway, this is going off topic, but it will hopefully be helpful for anyone who wants to know how long it takes to watch all episodes as that's usually the precursor to a Doctor Who marathon.
It's possible the asker wants to know physically how long it would take to view without interruption. A very approximate estimate based on the episode counts at Wikipedia work out as follows. Note that this does not take into account actual episode lengths, since some episodes ran more than their allotted time, and some ran less (there were classic-era episodes that ran only 18 and 19 minutes). I also ignore the question of what episodes still exist because all "wiped" episodes are widely available in audio form, so even if you're not able to watch them on TV you can still listen to them, so they can still be counted.
681 episodes x 25 minutes (1963-1984; 1986-89)
96 episodes x 45 minutes (1985; 2005-2008; 2010-October 2012): does not include Christmas specials. Does not take into account overruns - i.e. Episodes that run longer)
7 specials x 60 minutes (2005-2011)
1 special x 75 minutes (2007's "Voyage of the Damned")
1 special x 90 minutes (Five Doctors)
1 TV movie x 95 minutes (1996 TV movie)
So far, the math works out to 22,025 minutes, or 367.08 hours, or 15.3 days of continuous viewing.
But we need to add further estimates. To include the mini episodes broadcast or posted online since 2005, I say add approximately one hour (I think they can all be viewed in an hour). Add 2 additional hours to cover the animated stories Infinite Quest and Dreamland. Then I say let's add another 2 hours, roughly speaking, to cover episode overruns since 2005. But let us then subtract an hour, roughly speaking, to cover cases throughout the classic era where episodes tended to run less than 25 minutes (sometimes as little as 18 or 19 minutes. So that brings things up o 22,265 minutes, 371.08 hours, and just under 15 1/2 days. If someone wants to add up the individual episode running times for a more accurate number, feel free! Moral is - lots of Doctor Who out there!