Neural Radiance Fields (NeRF) have achieved huge success in effectively capturing and representing 3D objects and scenes. However, several factors have impeded its further proliferation as next-generation 3D media. To establish a ubiquitous presence in everyday media formats, such as images and videos, it is imperative to devise a solution that effectively fulfills three key objectives: fast encoding and decoding time, compact model sizes, and high-quality renderings. Despite significant advancements, a comprehensive algorithm that adequately addresses all objectives has yet to be fully realized. In this work, we present CodecNeRF, a neural codec for NeRF representations, consisting of a novel encoder and decoder architecture that can generate a NeRF representation in a single forward pass. Furthermore, inspired by the recent parameter-efficient finetuning approaches, we develop a novel finetuning method to efficiently adapt the generated NeRF representations to a new test instance, leading to high-quality image renderings and compact code sizes. The proposed CodecNeRF, a newly suggested encoding-decoding-finetuning pipeline for NeRF, achieved unprecedented compression performance of more than 150x and 20x reduction in encoding time while maintaining (or improving) the image quality on widely used 3D object datasets, such as ShapeNet and Objaverse.