I just want to ensure that I'm understanding you correctly. Typically a service area represents all of the accessible edges within a specified limit. For example, let's consider the below image. In this image I've created a network with four lines that extend exactly one mile in each direction from the center.
If I were to execute a one mile service area from the center of the network it would show that I can travel anywhere along the network.
Due to the simplicity of this network, I would assume in your case that the above solution returned a total of four miles. In your case I'd assume that you'd want something that would allow you to input something like "create a service area where the sum of the lengths of the edges traveled equals one". In this simplified case where all is equal I know that I'd need to travel 0.25 miles along each edge to get the out-of-the-box tools to meet this requirement in the simplest manner.
Is this what you're seeking, but in a more real world example where the solution will vary drastically? I don't believe there is anything out-of-the-box that would yield this solution, but if needed I could see how you could code in most of Esri's available APIs to determine this answer. Could you provide some more information about your business requirements (i.e. what you're hoping to gain from this analysis and why you've chosen this approach)? Also, do you already have an idea of what your needed solution would look like and how you're intending to handle the infinite number of possibilities that this solution would yield. For example, the following image would be just as correct as the above image if I only needed a result where the sum of the lengths of the edges traversed is one mile. Technically I could distribute the required total distance along any number of edges and the solution would be correct.