I had rather significant eczema as a child. I even used prescription medication to treat it for a time and then stopped as I was able to manage it effectively through other means as I got older. I had a lot of anxiety about it when I was applying to the academies and then again before I graduated as I went through my pre commissioning physicals. Fortunately, it has not been an issue for me thus far. I was given waivers for it by both USNA and USMA, and then later on I was given waivers for all the service communities that I could have gone post USNA. I'm now in flight school.
Some people will say that eczema is a huge dealbreaker and that NOBODY ever gets a waiver for it. That's not entirely true, it can happen, but everybody's situation is different. I have been extremely fortunate but I do know of others with eczema that have not had the same degree of luck.