Battery Draining - Sequoia 2002

I have a Toyota Sequoia (2002) which is really weird. It will be fine for a while and then suddenly it will have a flat battery. I thought I was leaving a door open but it worked fine today - I went shopping for 30 minutes and - BINGO - a shopping cart full of stuff and two kids.....no juice in the battery.
Is this the alternator or the battery? The car only has 44,000 miles on it :(

I love the car - but this is a BIG PAIN in the %%$#!!@.