For some reason (because it's a very new motherboard I find it strange), my motherboard has no eSATA connectors, but it has 8 SATA II connectors. My case also has nothing that would allow having an external eSATA connector. I'm wondering if there exists ways to make such a computer support external eSATA disks? Do there exist extension cards that support this? Do their exist connectors that can connect an eSATA disk to a SATA II slot on the motherboard? Will this have the 3 GBit/s bandwidth?

You do realize eSATA is just an alternate style connector for SATA, much like a USB B plug (the square USB plug you often plug into a printer) is just an alternate plug style for USB A?

All an eSATA does over a SATA connector is provide a little more robust physical plug. While a few motherboards have a single built in eSATA connector for convience, you normally get them by just sticking on a eSATA header card (same idea as USB headers, only even simplier in design as it's 100% pass through) that provides access to your internal SATA connectors.