Building a gaming PC for under $800 is absolutely doable in 2025, and the result is far more capable than any console at the same price point. The Ryzen 5 5600X on the AM4 platform gives you access to affordable DDR4 memory and mature, well-priced motherboards.
The Radeon RX 7600 delivers excellent 1080p performance and comes with 8GB of VRAM, which is enough for current games at medium-to-high settings. Pairing it with 16GB of DDR4-3200 keeps the budget tight without bottlenecking performance in any meaningful way.
This build is an ideal first PC or a solid choice for someone who games casually and doesn't need to max out every setting.