Excellent Advice already given. My 2¢ below.
Depends what you want to do with the rifle.
For accuracy and LR, being cheap doesn't help in most cases, but you could be the proverbial blind squirrel that finds the nut.
Of the actions I have or have used*, in this order of preferred donors:
- Tikka
- Howa 1500 / Vanguard
- *Browning (never felt a bad one)
- Winchester (caveat emptor, some are awful)
- CZ 550
- Remington
- Savage
- Ruger
- Marlin
Didn't list Weatherby Mark V or Accumark, or any Sako because (IMO) those rifles are worth more unaltered unless the barrel is toast. No comment on Bergara because I've never owned or shot one. With any of the actions listed above, you can chunk up some money quick "truing" them to where the difference between the action and a custom like a Defiance, Zermatt or BAT *might* be as little as $500. In the life of a rifle that you love to shoot, that $500 is
NOTHING.
Myself, I have rebarreled and wildcatted Remington and Savage.
[Edit to add] Unless they were shot out, I would never rebarrel a Browning, Winchester or CZ. To me that would be sacrilege. [/edit]
Cheapest DIY custom build out the door is Savage, but YMMV. Reasons why you would NOT want a Savage: less after market options, especially stocks, and the extractor + ejector leaves a lot to be desired. Less resale too because of snobbish attitude about Savage.
The last Savage action I rebarreled, the smith faced off the bolt face, and said there was less than 1 mil difference across the action face.
Remington actions are stronger than Savage actions. Remington bolts are stronger. A great deal more after-market parts availability for Rem. But, sometimes at a cost -- need to change your bolt face, on a Rem, BOHICA.
Savage is easier in that respect because of the modular bolt design - easier to adjust your timing too.
If this is your dream custom rifle and might be the only one you'll build consider time and expense.
If you're on a budget, consider the time you'll wait for a barrel and in most cases, smith services, and save your money for the custom action rather than "getting started". To wit, my first two custom rifles were built on a budget, and each took > 18 months from start to finish. My first barrel took 6 months to arrive. YMMV
If you have friends with rifles built on premium actions, ask to shoot them. The difference is telling. If this is a rifle you'll use in competition, you want a premium action. If it's something you'll pull out twice a year, for sight-in and hunting, an OTS action would do just great.