There are two ways to get the data from the DAT playter to the PC. You can transfer the digital file directly or you can let the player convert to analog by playing the tape and capture the analog out and convert it to digital.
The digital transfer would be the best, but I do not know what format the DAT files are in. As Raymonda says there are mulitple optons for digital recorders that take in digital signals. They use S/PDIF inputs, so you would need a S/PDIF output from your player.
Capturing the analog means you go through a digital to analog conversion and then an analog to digital conversion.
To use the analog option, let your DAT player play the tapes and then capture the analog output with a recording system. All you need is a PC with a analog input and some software. The A to D converter in the PC may not be the greatest quality, but it is a start. An external analog to digital converter is also an option, although that will cost more. They are available at multiple prices from companies like m audio, Tascam, Korg. A decent one is availabe for a few hundred dollars. You can capture the audio with recording software like Audacity or Vinyl Studio.
Vinyl Studio is specifically meant for recording from vinyl or the analog output from a CD player. I use it for vinyl and like it because you can record the whole album (tape) and then it can split the tracks and has a lookup option to get the tag data. Few other audio editors have this lookup feature and the price is right at $29. Trial available.
A direct digital to digital transfer would be best, depending on the format in the DAT and your digital output option. Recording the analog output is an easy option if you have a computer with analog inputs.
Once you have the digital file, then you need either a dedicated music server or a PC with player software and a DAC. The music server is easier to implement but the PC approach is more flexible. For a windows PC, you can use foobar (free) or JRIver ($50) for a player. iTunes or Windows Media Player are options, but the quality is generally not as good. JRiver and Audiovarna are optons for Apple. You also need a DAC, either in a receiver or intergrated amp or stand alone. You can get a decent usb DAC starting DAC for $500 or less, depending on the rest of your system.
If you want a stand alone music server, you might want to look at the Sony HAP-S1. It copies digital files from your PC and then acts as a stand along system. Price $995.
Hope that helps. There are lots of options. Some of it depends on your computer expertise. But there are certainly mainstream options for much less than the Cocktail Audio.