Is it possible to form an 8 carbon ring using a Diels-Alder reaction?
You can obviously use Diels-Alder Reaction for the synthesis of the desired product but not so directly. But Diels-Alder will give an optimal yield when the ring formed is Six-Membered. After the formation of that Six-Membered product, you have to increase the no. of carbon atoms in the carbon-chain by some way, and the yield of your product will be also better if you no. of steps as less as possible. 

I will not suggest you to directly add 1,3-butadiene to acetylene because there will be a chance of aromatisation, and then you cannot do anything, as to break the aromatic rings are highly unfeasible. Instead I can suggest you the following steps:


First you partially reduce the acetylene to ethene, and then perform Diels-Alder reaction with butadiene. You will get cyclohexene.
In the second step, you perform ozonolysis, and then to the resulting compound, perform  Wittig reaction with excess reagents. You will get octa-1,7-diene.
Lastly, you can perform hydroboration, oxidation to get less substituted diol, and then oxidise that to acid by any strong oxidising agent like $\ce{KMnO4}$ or $\ce{K2Cr2O7}$ in slightly acidic medium. 


Thus, you will get your desired product. The path of reaction is shown below: