This will work for Linq to Objects, but may not be supported by Linq to Entities:
var strings = _dataContext.Context.Detail.AsQueryable()
.Where(n => n.DetailNr == 6)
.Select(n => new [] {n.Str1, n.Str2, n.Str3, n.Str4});
.SelectMany(s => s);
var result= _dataContext.Context.String.AsQueryable()
.Where(s => strings.Contains(s)).ToList();
You may need to add as AsEnumerable
to hyrdate the list.