Well, the variance does increase away from the data, just maybe not as quickly as you'd expect. From what I understood, this depends on the details of your network (the activation function, in particular) and the training parameters. Ultimately, those factors decide what kind of Gaussian process kernel you're effectively using, but the connection between the two isn't straightforward (and an active area of research, as far as I could figure out).

Also, Gaussian processes do not necessarily produce arbitrarily large variance away from the data from what I've seen. If you take a simple example from the documentation of Predict, you can quite easily get a constant error band away from the data:

data = {-1.2 -> 1.2, 1.4 -> 1.4, 3.1 -> 1.8, 4.5 -> 1.6}; p = 
 Predict[data, Method -> "GaussianProcess"];
Show[Plot[{p[x],
   p[x] + StandardDeviation[p[x, "Distribution"]], 
   p[x] - StandardDeviation[p[x, "Distribution"]]},
  {x, -5, 10},
  PlotStyle -> {Blue, Gray, Gray},
  Filling -> {2 -> {3}},
  Exclusions -> False,
  PerformanceGoal -> "Speed", 
  PlotLegends -> {"Prediction", "Confidence Interval"}], 
 ListPlot[List @@@ data, PlotStyle -> Red, PlotLegends -> {"Data"}]]

I think it depends on how you do your GP regression in this case and what kernel you use. If you make a point estimate for the covariance length scale, I think you end up with constant prediction variance far from your data (since basically none of the data points are correlated with the prediction point). If you do a proper Bayesian inference where you integrate over the posterior distribution of your length scale, it will be different since you'll have contributions from very long length scales.

In the blog post by Yarin he also shows an example of GP regression with a square exponential where the error bands become constant.