We consider light, high-absorbance, low-reflectance, electrically large layered sheet structures composed of thin carbon nanotube films. Such structures can be utilized in electromagnetic absorption and shielding applications in the X-band. They are of increasing interest in sensor-enabling technologies, stealth systems, and EMI shielding of electronic components. Especially in aerospace, this is crucial, as sensors are integral to aerospace engineering, enhancing the safety, efficiency, and performance of aircraft and spacecraft. To that end, sheets with carbon nanotube films embedded in a glass fiber polymer matrix are fabricated. The films have a thickness of around 70 μm. As shown, they cause a significant attenuation of the electromagnetic field. For shielding applications, a single-film sheet structure with total thickness of 1.65 mm presents an attenuation of around 25 dB in the transmission coefficient, while the attenuation can reach 37 dB for a two-film sheet structure with thickness of 1.8 mm. Shielding effectiveness performance is found to be greater than 35 dB for the two-film sheet structure. For applications requiring both high shielding and absorption, a two-layered structure with a thickness of 4.65 mm has been designed. The absorption, represented by the Loss Factor, is calculated to achieve values greater than 90%. The simulation results show good agreement with the measured data. The findings demonstrate a promising structure for materials suitable for sensor housings and smart electromagnetic environments where the suppression of electromagnetic interference is critical. In conclusion, the addition of carbon nanotube films, even at micrometer thicknesses, within a glass fiber polymer matrix significantly enhances both electromagnetic shielding and absorption performance.