For common, affordable, non-restricted options you have the 7.62x54R or the 7.62x39, both of which use .311" or 7.65mm diameter bullets, similar to a 303 British, but if you handload it's wise to actually check your barrel dimensions, again like a 303.
The 7.62x54R has the most muzzle energy, but usually uses heavier bullets with lower muzzle velocity than a 308 or modern 30-06. Because it is rimmed, be gentle when you load your magazine.
In this chambering, likely choices are Mosin-Nagant, SVT, and Dragunov rifles.