The network will keep working until your DHCP leases expire. After the leases expire devices may switch to RFC 3927 addresses. But those addresses are not predictable so you'd have to rely on MDNS to find them, and they are unlikely to work between a given pair of devices until both have switched from DHCP assigned addresses to RFC 3927 addresses.
On the DHCP server you can increase the lease time. If this network is behind a NAT you will likely have plenty of addresses and can configure a very high lease time. But it will only work for clients which already had an address.
You probably want to configure the NAS itself with a static IP address. (That will mostly rule out RFC 3927 addresses as fallback, but that wasn't much of a fallback in the first place.)
Unless your clients connect to the NAS by IP address or have the NAS in their hosts file, they will need DNS to find the NAS. The TTL of the DNS record decides for how long they will be able to find the NAS after DNS goes down. But keep in mind that where DHCP is commonly refreshed after half the lease time, DNS records are usually only refreshed once they have already expired. So even with a high TTL the DNS records may expire. And only clients which had cached the name will be able to use it. It is possible that each client machine has its own DNS cache, but it is not guaranteed. You could end up relying on application layer DNS caching, and restarting the application accessing the NAS may lose the cached DNS record.
An alternative I recommend considering if continued operation in case of loss of DHCP and DNS is a must is to use IPv6 link-local addresses. All modern operating system have them turned on by default and devices generally have at least one address that remains static regardless of changes in your network configuration.