What is the lowest temperature possible (of water) to cook meat in?
The answer depends on the type of meat as well as the thickness. There are different things you need to worry about killing in different kinds of meat, and it takes a different amount of time for everything to get hot enough for long enough depending on the thickness.

Note that you generally don't want to put the meat straight in the water, since you'll lose flavor into the water (unless your goal is to make broth or soup); you'll want to put it in a sealed bag in the water. This method of cooking is called sous vide. (Yes, that means "under vacuum", but you don't actually need vacuum-sealed bags; airtight with most of the air removed will suffice.)

The best single resource I know for this is Douglas Baldwin's "A Practical Guide to Sous Vide Cooking". There are tables for fish/shellfish, poultry, and beef, pork and lamb, as well as a lot of other great information. To give you a rough idea, the temperatures in the tables go down to around 55C, and at that temperature, the times cover a huge range, from 2 hours for thin (5mm) beef, pork, or lamb, to 9.25 hours for thick (70mm) fatty fish. 

That 55C is the lowest safe temperature I know of. The government-published pasteurization time tables don't go below that. It's possible lower temperatures can work too, but you'd have to start digging into published papers.

Note that there may also be thickness/size bounds. Those tables only go up to 70mm so that's all I'm confident about. It's possible you can go even thicker, but I don't think you can go on forever. Eventually things will be sufficiently thick that the amount of time spent at too-low temperature in the center of the meat before it's totally heated up can't be compensated for by time holding it at a higher temperature once it's fully heated. This would be especially true if you try to cook anything that's not totally solid, since there's more contamination on surfaces than inside meat. For example, cooking a whole chicken like this doesn't sound like a great idea.