The advent of nanocarbons, from single- and multiple-walled nanotubes to nanohorns, avails model studies of confined molecules on the nanoscale. Water encapsulated inside the quasi-one-dimensional channels of these materials is expected to exhibit anomalous behavior due to the unique geometry of nanotubes and the weak interaction between the water molecules and the carbon atoms. We have employed neutron small-to-wide angle diffraction, quasielastic and inelastic scattering in conjunction with molecular-dynamics simulations to characterize the structures and dynamics of water adsorbed in open-ended single- and double-walled nanotubes over a wide range of spatial and temporal scales. We find that a square-ice sheet wrapped next to the inner nanotube wall and a water chain in the interior are the key structural elements of nanotube-confined water/ice. This configuration results in a hydrogen-bond connectivity that markedly differs from that in bulk water. This significantly softened hydrogen-bond network manifests in strong energy shifts of the observed and simulated inter- and intra-molecular vibrations. The very large mean-square displacement of hydrogen atoms observed experimentally and the strong anharmonicity inferred from simulations explain the fluid-like behavior at temperatures far below the freezing point of normal water.