Roast beef is what you are looking for.
Let me tell you up front, though, that without an expensive meat slicing machine, you'll never match exactly what you get at Arby's but you might come close.
Making good roast beef is not something you'll master overnight but if you put the time in and make a few, with a little trial and error you should be able to create something very Arbylike
First of all, get yourself a $25 (or less) digital probe thermometer from somewhere like target, walmart or kmart. The kind you are looking for has the long wire so the probe can stay in the meat during cooking. Then get yourself a rump roast - look for a cut with the most white flecks in it (fattiest). After you've mastered a rump roast you might want to try an eye of round (more traditional). Because it's such a leaner cut, it's quite a bit more difficult to get a moist and flavorful roast.
Prep
Depending on the store where I get if from, sometimes I rinse off my roast with water to clean it. Other times I just blot it with a paper towel. Whichever you do, make sure it's dry going into the oven. Insert the probe thermometer sideways into the meat so the point is in the thickest part (middle). If you have a rack for your roasting pan, use it, if not, it's not the end of the world. Put the roast in a preheated oven.
Roasting
What temp for the oven? The traditional method is to roast at 325 degrees until your thermometer reads your desired temp (140 degrees for medium rare). That will give you a okay roast beef. Some people will tell you to sear the meat first to 'seal' in the juices. That is a myth. Seared meat, as it turns out, loses more moisture than unseared meat.
If you , like me, are striving for the ultimate roast beef, I'd recommend a slightly different tack. I've been seeing more and recipes from chefs instructing an initial heat of 375 (for browning and food safety) but then going down to 250 for a very slow roasting. This method will give you a moister, more delicious roast. I've never done it with beef but I have done it with pork and the results have been amazing. How long at 375? Well, I have to admit I've been doing mine by eye. The next time I roast I'll pay closer attention to the thermometer but if I had to make a good guess, I'd say roast at 375 until the probe thermometer hits 90 then lower to 250 until you hit a 135 degree internal temp. As far as the final temp is concerned that's really a matter of taste. When you take the roast out of the oven it will continue to cook another 5 to 10 degrees depending on the size. 140 is supposed to be medium rare but everyone you talk to seems to have a different opinion on internal roasting temperatures. 140 is way too pink for me personally. If it is too pink/red, remember you will be rewarming it after you cool it/cut it. Arbies probably takes theirs to 150 or maybe even higher. You'll probably need to make roast beef a couple of times until you hit the exact color you want.
One important thing to keep in mind during roasting is to keep an eye on your pan juices. If you see them starting to burn/get black add a few tablespoons of water. Generally you don't need to inspect for burning until about 1/3 through the cooking time. Once out of the oven, pour the pan juices into a separate cup and chill. If you don't want to monitor your pan juices you can substitute them later with water, but your sandwich won't be as good.
Cutting
Getting really thin slices from a straight from the oven roast beef is impossible. Let the roast cool for about 30 minutes or so and then refrigerate overnight. After fully chilling the roast will be a lot easier to slice. Some people tie up their roasts with butcher twine so it's easier to cut and other people partially freeze their roast beef. I just use a very sharp 8" chef's knife. Cut thin slices against the grain. If you have a friend who's a butcher, access to a meat slicer would be ideal.
Rewarming
Rewarming is probably the most difficult part for me. I've tried warming in some hot water (bad idea) then in a microwave (worse idea) and finally came upon the oven technique which works well as long as you watch it carefully. First get your bun ready/split on a piece of foil and near the oven. Preheat oven to 275. Microwave your chilled roasting juices a few seconds so they are fully liquid. Line a cookie sheet with foil and then coat the sheet lightly with some of the juices (or a spritzing of water). Arrange your slices in a single layer and then lightly coat the tops with the same juice (or water). If you were able to achieve paper thin slices, a single layer isn't necessary, I just clump them in a thin layer. Now here's the tricky part, depending on the thickness of slice you were able to achieve, you want to bake them at 275 degrees just long enough to warm them. If by chance they were a little too pink for your taste to begin with, you can bake them a little longer, but pay very close attention as they turn grey/tough in a heartbeat. Once they are warm enough, immediate transfer them to your open bun and wrap with foil. In about 15 minutes you'll have a wonderful fast food style roast beef sandwich.
