You'd have to start with a dedicated or at least somewhat purpose-built rifle. Of course you can shoot any rifle off of a table and call it "benchrest" but it's not...... Bench Rest means something. Bench Rest is like the Top Fuel division of rifle shooting, ACCURACY rules at the expense of all else. You could also build a hotrod to drive around and call it "racing" but a real race car is purpose-built, not road legal. True benchrest is the pinnacle of accuracy and requires enough specialized equipment that the rifles used are somewhat limited for real world usage. I'm not saying you won't hunt with a BR rifle but they gener'ly don't hunt too good
First of all go to IBS and NBRSA websites and check for matches within your driving range....even if you decide not to compete it's worthy IMO to go to a match just to
SEE the stuff.
Next you have to understand that except for a couple weird exceptions you WILL NOT buy a Bench Rest capable rifle from any manufacturer. One specialized rifle that technically _could_ conceivably place in a small BR match would be a 22PPC USA and 6PPC USA rifle as built by Sako for about 6yrs in the early 80's, if the barrel is still good. I sold one a couple yrs ago that was still easily shooting 1/4" groups, enough to occasionally get you on the board at a BR match. There have been a few tight-necked Coopers that exhibited decent accuracy, and a few other custom-shop rifles that through serendipity came out "tight" or "straight" but these are luck of the draw.
For a rifle to actually exhibit the accuracy required for any type of bench rest shooting it must be built to tighter specifications than are possible in mass produced guns. Even the Savage rifles sold for 1000.00 and MARKED "Bench Rest" don't shoot well enough to BE actual Bench Rest capable.
You want to get a taste of REAL accuracy my advice is to go to the classifieds here on this site or over to Shooters Corner, 'The List' and buy a used setup. For raw intrinsic jaw-dropping accuracy at 100-200-300yds get a 6PPC. For anything capable of winning accuracy at longer range my advice is to pay a little more for a proven winner. Accurate long-range rigs are about 20:1 harder to find than accurate PPC's. Also much harder to gear up and learn to load for....
Also, keep hanging out here, keep asking questions, clarify your wants/needs........... this is the one site in the world where the actual record holders, gun builders, barrel makers, stock makers, accessory makers hang out.
enjoy
al