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Without NAT, you can't use private IPv4 addresses in your network (with Internet connectivity). You'll need to get a sufficient IP address range from your ISP, at least one address for each device with Internet access.
You'll still need a router but it could route public-to-public without any need of NAT. Note that this is the standard way to connect IPv6.
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There is absolutely no need to use NAT or private addresses to connect a network to the Internet. NAT was only designed as a short term solution to work around the lack of IPv4 addresses (and yes: NAT is old. People noticed that we are running out of vintage IP (IPv4) ages ago).
There are two different scenarios, it doesn't matter if they are vintage IP or ...
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NAT is only required if you have overlapping addressing, or you are trying to connect a privately addressed network to the public Internet (which is really a form of overlapping addressing because other sites may use the same Private addressing).
The Private IPv4 address ranges were selected to allow anyone to use those addresses on their own networks, and ...
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There are two main differences between a "with NAT" and a "Without NAT" scenario.
With NAT you can use anything from the huge range of private IP addresses, without it you are limited to the public addresses you are allocated.
Your ISP must allocate a block of IPs and route it to your router.
How hard this will be depends on how big a block you want and ...
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