>Is it at all possible to get (or make) a motherboard which supports both 64bit CPUs and ISA with DMA transfers?
Yes
>So my question is, what are the hardware limitations that make 64bit and DMA impossible or infeasible to mix?
None unless imposed by the manufacturer that put the IOMMU on the board in agreement with the chipmaker for the CPU like jewtel. Most limitations are made in software like for windows.
>Is this a CPU problem or a motherboard design problem?
Motherboard problem https://archive.fo/xCBbx .
>As in, is can 64bit code do old fashioned 8 or 16bit era DMA transfers, using the equivalent of an 8237 chip?
Yes it can https://archive.fo/AS31f if the IOMMU supports it. Why is it you want to do this though?