This car came with a Group 94 sized battery 640 CCA. That is a long, low profile battery so it fits under the trunk floor/tray.
Group 94 is fairly unusual. Not all vendors carry it and if there are not many BMWs in your town, any 94 that you find may have been sitting on the shelf for a long time. Batteries are shipped wet these days, so it is better to find a fresh one, not a maintenance free battery that has been sitting on the shelf partially charged for months.
There are only three companies in the world that make all the lead/acid automotive batteries, and one company that makes the machinery that manufactures the batteries, so there is not as much difference between different brands of batteries as the vendors of expensive batteries would have you think. Some brands use recycled lead and some batteries are made in plants in third world countries, so you might want to be cautious of those.
A group 94 is 6.9 inches wide and 7.5 inches high and 12.7 inches long. Your tray is longer than this if you remove the plastic spacer from the end of the tray, and some folks report installing a Group 49 which is just over 14" long. Wal Mart sells a Group 49 for $90, but I did not try it because it looked to me like it would be too big.
I ended up buying a Group 48 from NAPA for $105 for the wife's '04 330. A group 48 is the right width but is smaller in height and length. You have to put a spacer on top of it so the battery bracket holds it securely. A Group 48 is a common size so it should be fresh.
This NAPA battery is rated at 700 CCA compared to 640 CCA for the original battery, but something is fishy. For a lead/acid battery, the CCA is roughly proportional to the weight of the lead in the battery. This new NAPA battery weighs about half what the original BMW battery weighed. I like carrying around less weight, but either my NAPA battery is mismarked, or someone has a creative way of calculating CCA. When my 325 needs a battery, I plan to move the NAPA to my 325 commuter car and put a better (heavier) battery in the wife's 330.
On an '01, you will loose your radio code and perhaps the presets (?) and perhaps the seat memory if you have one. Nothing serious. You can avoid this by taking another car and connecting jumper cables to the connections under the hood while you are changing the battery. I swapped the wife's '04 330 in the NAPA parking lot with no other power source and I lost nothing. Cables were disconnected for less than a minute.