Possible to build good Gaming PC for €550-ish?

It will probably be able to play games for many years to come, you'll just have to start turning the details down.
New games don't progress as fast as they used to as most developers have to keep an eye on the console world. They'll want they games to run on consoles to give them more people to sell to. A new PC like that would be several times the power of the console and with the rumoured specifications of the new xbox, a 6870 would still be more powerful than that. Consoles have the advantage of having games being very tightly coded for them to take advantage of the hardware to the maximum whereas a PC has a lot more to juggle with like a full operating system.

The best way to stay to save you from having a completely obsolete computer in the near future is to be careful in selecting your components for future upgrades. The worst option is to get something like a Dell which may not support any upgrades as you might need particular motherboard type and the PSU mightn't be able to take on a graphics card. When you build yourself, all the components are standardised.

-A case like the Bitfenix Merc (I'd go with the Beta instead though if it was me), has room for extra hard drives or fans for the future. Motherboard fittings are standard if you wanted to replace that too.
-The H61 motherboard should be able to support all 1155 CPUs of this generation and the new Ivy Bridges allowing you to upgrade to a quad core or whatever. If you really wanted to keep your options open, you could get a Z68 motherboard which would allow you to overclock any "K" series CPU. However, it would double the cost of the motherboard. More expensive motherboards allow for multiple graphics cards.
-500GB hard drive should last you a while and it's a good model, but you can add in more hard drives later.
-8GB is a lot of RAM for now. Most people won't even use 4GB. That's a lower end motherboard so it only has 2 slots but many have 4.
-The 6870 is a good GPU and the most important thing for games. That and the CPU would be the first thing that you'd be looking at if you needed upgrading later on. The 450W PSU would be enough to support that or equivalent cards.

To be honest I don't think you have anything to worry about with Gpus like 6870, if you support CF then even better as there should be few of them on adverts.

Pc graphics are pushed back by consoles these days. Developers won't be bothered to make 2 separate games for console and pc. They just make console version and then port it to pc, so that's why pc games became less demanding. Ofc we got awesome exclusives, but of they will scale very well with hardware.
If rumors are true about Xbox360 + 90, then pc gamers can't relax with 6870++ Gpus. Keep in mind: upgrade bug will bite you way sooner then outdated hardware bug.

Ok....Give us pc newbs the skinny! Taking the upgrade bug out of the equation, how long will these €500-700 PCs last us? Like in 7 years, without upgrading, can i still play the 2019 games in the same quality/res as today's games? Or will computer benchmarks be too high?