When John Webb takes his family to California, Washington and Oregon for vacation from his home in Dutchess County, N.Y., they don’t fly or drive. They travel hooked to the back of an Amtrak train in a 1950s-era private railcar that Mr. Webb spent three years and nearly $1 million restoring.

Traveling across the country by private rail takes a lot of planning, requiring connections to various trains and layovers. But Mr. Webb, 35, says he and his 10-year-old son and 8-year-old daughter prefer it to their family’s second home...