And let me emphasize that's still suburban, not rural. I live 8 minutes from a good grocery store, 30 minutes from a city of 46,000 people. Small roads aren't paved here (the nearby highway is, and the in-town streets are), and the 20 mph road maintained by nobody else but a couple of residents with tractors is 400 feet from the house.
(And the trade-off for this level of remoteness, for those wondering, is I have about 200 square miles of public lands walkable from the house without crossing a highway. Ever taken a two-hour walk from your house, with your dogs off leash, without seeing another human?)
That's really a question of pick your climate. Colorado's mountains get a lot of snow in the winter, Arizona's deserts never get really cold, Utah gets both snow and hot summers. All of Southwest has quiet spots if you look for them.
I'm an hour south of Colorado's mountain ranges, in a little pocket of microclimate where most storm clouds sweep past you on both sides, where winter still exists but where you don't need to shovel snow, and where summer days stay below 100 F. We looked at houses in 3 states before settling on this.
Look into Arizona, New Mexico, Utah (Moab is an outdoor sports mecca), California (around highway 395, and on the coast outside of big cities, are some pretty areas), Colorado (if you love skiing etc), maybe Texas (e.g. "Hill Country").
And if you want more free space, move to Wyoming or Montana (but good luck with the groceries).
We spent just under 3 years roaming the Western US with a motorhome. I've seen a lot of natural beauty, you just have to leave the cities and it's there. The rest is mostly a matter of what climate, plant life, etc you prefer, and how close do you want to live to a good hospital, groceries, UPS & mail delivery, etc.
(And the trade-off for this level of remoteness, for those wondering, is I have about 200 square miles of public lands walkable from the house without crossing a highway. Ever taken a two-hour walk from your house, with your dogs off leash, without seeing another human?)