Aperture-synthesis and single-dish (sub-) millimeter molecular-line and continuum observations reveal in great detail the envelope structure of deeply embedded young stellar objects (SMM 1 = FIRS 1, SMM 2, SMM 3, SMM 4) in the densely star-forming Serpens Molecular Cloud. SMM 1, 3, and 4 show partially resolved (>2''=800 AU) continuum emission in the beam of the Owens Valley Millimeter Array at λ=3.4-1.4 mm. The continuum visibilities accurately constrain the density structure in the envelopes, which can be described by a radial power law with slope -2.0±0.5 on scales of 300 to 8000 AU. Inferred envelope masses within a radius of 8000 AU are 8.7, 3.0, and 5.3 M for SMM 1, 3, and 4, respectively. A point source with 20%-30% of the total flux at 1.1 mm is required to fit the observations on long baselines, corresponding to warm envelope material within ~100 AU or a circumstellar disk. No continuum emission is detected interferometrically toward SMM 2, corresponding to an upper limit of 0.2 M assuming T_d=24 K. The lack of any compact dust emission suggests that the SMM 2 core does not contain a central protostar. Aperture-synthesis observations of the ¹³CO, C¹⁸O, HCO⁺, H¹³CO⁺, HCN, H¹³CN, N₂H⁺ 1-0, SiO 2-1, and SO 2₂-1₁ transitions reveal compact emission toward SMM 1, 3, and 4. SMM 2 shows only a number of clumps scattered throughout the primary field of view, supporting the conclusion that this core does not contain a central star. The compact molecular emission around SMM 1, 3, and 4 traces 5''-10'' (2000-4000 AU) diameter cores that correspond to the densest regions of the envelopes, as well as material directly associated with the molecular outflow. Especially prominent are the optically thick HCN and HCO⁺ lines that show up brightly along the walls of the outflow cavities. SO and SiO trace shocked material, where their abundances may be enhanced by 1-2 orders of magnitude over dark-cloud values. A total of 31 molecular transitions have been observed with the James Clerk Maxwell and Caltech Submillimeter telescopes in the 230, 345, 490, and 690 GHz atmospheric windows toward all four sources, containing, among others, lines of CO, HCO⁺, HCN, H₂CO, SiO, SO, and their isotopomers. These lines show 20-30 km s^-1 wide line wings, deep and narrow (1-2 km s^-1) self-absorption, and 2-3 km s^-1 FWHM line cores. The presence of highly excited lines like ¹²CO 4-3 and 6-5, ¹³CO 6-5, and several H₂CO transitions indicates the presence of material with temperatures 100 K. Monte Carlo calculations of the molecular excitation and line transfer show that the envelope model derived from the dust emission can successfully reproduce the observed line intensities. The depletion of CO in the cold gas is modest compared to values inferred in objects like NGC 1333 IRAS 4, suggesting that the phase of large depletions through the entire envelope is short lived and may be influenced by the local star formation density. Emission in high-excitation lines of CO and H₂CO requires the presence of a small amount of ~100 K material, comprising less than 1% of the total envelope mass and probably associated with the outflow or the innermost region of the envelope. The derived molecular abundances in the warm (T_kin>20 K) envelope are similar to those found toward other class 0 YSOs like IRAS 16293-2422, though some species appear enhanced toward SMM 1. Taken together, the presented observations and analysis provide the first comprehensive view of the physical and chemical structure of the envelopes of deeply embedded young stellar objects in a clustered environment on scales between 1000 and 10,000 AU.