Three-dimensional arrays have the ability to localize sources anywhere in the spatial domain without any ambiguity. Among these arrays, the spherical microphone array (SMA) has gained widespread usage in acoustic source localization and beamforming. However, SMAs are bulky and in many applications with space and power constraints, it is undesirable to use an SMA. To deal with this issue, arrays with microphones placed only in a sector of a sphere have been developed along with various techniques for localizing far-field sources in the spherical sector harmonics (S²H) domain. This work addresses near-field acoustic localization and beamforming using a spherical sector microphone array. We present the representation of spherical waves from a point source in the S²H domain using the orthonormal S²H basis functions. Then, using the representation, we develop an array model for when a spherical sector array is placed in a wavefield created by multiple near-field sources in the S²H domain. Using the developed array model, two algorithms are proposed for the joint estimation of the range, elevation and azimuth locations of near-field sources, namely NF-S²H-MUSIC and NF-S²H-MVDR. Further, a near-field beamforming algorithm capable of radial and angular filtering in the S²H domain is also presented. Finally, we present the Cramer-Rao Bound (CRB) in the S²H domain for near-field sources. The performances of the proposed algorithms are assessed using extensive localization and beamforming simulations.