Someone correct me if I’m wrong but:

DIY Anycast is probably beyond most people’s reach, as you need to deal with BGP directly.

One cool trick is using GeoDNS to route the same domain to a different IP depending on the location of the user, but there are some caveats of course due to caching and TTL.

EDIT: Back to Anycast, there are also some providers who allow you BGP configuration, like those: https://www.virtua.cloud/features/your-ip-space - https://us.ovhcloud.com/network/byoip - https://docs.hetzner.com/robot/colocation/pricing/ ... However you still need to get the IPs by yourself, by dealing with your Regional Registry (RIPE in my case, in Europe)

You could start using DNS Traffic Shaping where DNS server looks at IP making the request and returns the IP of closest server.

Azure/AWS/GCP all have solutions for this and does not require you to use their services. There are probably other DNS providers that can do it as well.

Cloudflare can also do this as well but it's probably more expensive than DNS.

To get anycast working, you need BGP, and to get it working well, I think you need a good understanding of BGP and a lot of points of presence and well connected at each. BGP's default metric of distance is number of networks traversed, which does funny things.

Say you're in city A where you use transit provider 1 and city B where you use transit provider 2. If a user is in city B and their ISP is only connected to transit provider 1, BGP says deliver your traffic to city A, because then traffic doesn't leave transit provider 1 until it hits your network. So for every transit network you use, you really want to connect to it at all your PoPs, and you probably want to connect to as many transit networks as feasible. If you're already doing multihoming at many sites, it's something to consider; if not, it's probably a whole lot of headache.

GeoDNS as others suggested is a good option. Plenty of providers out there, it's not perfect, but it's alright.

Less so for web browsers, but you can also direct users to specific servers. Sample performance for each /24 and /48 and send users to the best server based on the statistics, use IP location as a fallback source of info. Etc. Not great for simple websites, more useful for things with interaction and to reduce the time it takes for tcp slow start (and similar) to reach the available bandwidth.