Lets start with your stated question:
Is it possible to have 60 hosts on a 255.255.255.248 network?
As you may have expected, the answer is no.
@null's answer to the question you asked is correct and I do not understand why it was downvoted. With a /29 you have (32-29=) 3 bits for your own local networking. That is 23 addresses. You want one of those as broadcast adress, one as network adress *1 and one for your gateway. That leaves you with 5 IPs which can communicate directly with the internet.
The keyword here is directly.
In a perfect world you would have more than enough IP addresses to give each device their own IP. And you would use on of them for a firewall which you would put between the gateway and your internal network.
Sadly it is not a perfect world.
Free IP v4 addresses are getting scarce. There are two ways around that:
- Do not use IP v4 but switch to IP v6
- Use an ugly hack called NAT/PAT.
The latter is probably the easiest way to get all your 60 devices connected to a local LAN with non-public IPs
*1: You might get away with not using the last. But it might break compatability with a lot of devices and it will give you headache when people assume that both .0 and .255 (or rather, in your case .192 and .199) are used in the traditional way.
*2Please make sure you use non public IPs. E.g. 172.16.130.x. See RFC1918 for more information on these IPs. And pick a range which is not commonly used. That will prevent problems later on when people start trying different NAT area's to one