Neural Network for Multiple Output Regression
What you are describing is a normal multidimensional linear regression. This type of problem is normally addressed with a feed-forward network, either MLP or any other architecture that suits the nature of the problem. 

Any neural network framework is able to do something like that.

The key to do that is to remember that the last layer should have linear activations (i.e. no activation at all).

As per your requirements, the shape of the input layer would be a vector (34,) and the output (8,).

Update: the usual loss function used for regression problems is mean squared error (MSE). Here's an example of multidimensional regression using Keras; the network is not an MLP but it should be Ok to illustrate the idea.