You haven't given the model number, but could be....
1) Faulty Capstan Motor bearings will certainly give the symptoms you've described. - Captan motor = expensive.
2) On some of the older models, I remember I often came across the take-up/half load tension arm sticking due to dried up lubricant.
(This is the small guide post that pulls the tape across the fixed audio/control head and capstan).
Without a cassette loaded the arm although under tension of a spring, should swing with little resistance. That can get stuck.
If it is that, clean the post that it swings on with an isopropyl alcohol product, (eg. I use Colclene TF, head cleaning fluid), then relubricate with a tiny, tiny drop of fine lubricant such as FLoil. Don't attempt to lubricate any other part of the mechanism.
3) Tape loading arms not completing travel, especially the supply loading arm. These two arms slide either side of the cylinder drum during loading and must meet their end stops. I've come across some JVC's and many other units where the loading arm has stopped just fractionally short. In which case a gear tooth on the arm may be damaged, or the arm sticking.
In all the above the video would automatically unload the cassette and eject it after a few seconds.
Ensure the product is unplugged if you check any of the above mechanical problems, as dangerous voltages exist inside the unit.
Cobweb, October 2005