Large portions of the endoplasmic reticulum (ER) in eukaryotic cells are organized as dynamic networks whose segments are connected by three-way junctions. Here we show that ER junctions move subdiffusively with signatures of fractional Brownian motion and a strong dependence on the cytoskeleton's integrity: The time-averaged mean square displacement scales as 〈r^{2}(τ)〉_{t}∼τ^{α} with α≈0.5 in untreated cells and α≈0.3 when disrupting microtubules, with successive steps being anticorrelated in both cases. We explain our observations by considering ER junctions to move like monomers in (semi)flexible polymer segments immersed in a viscoelastic environment. We also report that ER networks have a nontrivial fractal dimension d_{f}≈1.6 on mesoscopic scales and we provide evidence that the organelle's dynamics is governed by fractons.