This article considers linear processes with values in a separable Hilbert space exhibiting long-range dependence. The scaling limits for the sample autocovariance operators at different time lags are investigated in the topology of their respective Hilbert spaces. Distinguishing two different regimes of long-range dependence, the limiting object is either a Hilbert space-valued Gaussian or a Hilbert space-valued non-Gaussian random variable. The latter can be represented as a unitary transformation of double Wiener-Itô integrals with sample paths in a function space. This work is the first to show weak convergence to such double stochastic integrals with sample paths in infinite dimensions. The result generalizes the well known convergence to a Hermite process in finite dimensions, introducing a new domain of attraction for probability measures in Hilbert spaces. The key technical contributions include the introduction of double Wiener-Itô integrals with values in a function space and with dependent integrators, as well as establishing sufficient conditions for their existence as limits of sample autocovariance operators.