How to build an accurate oscillator around 1 kHz?
For a stable, low distortion sine wave of a fixed frequency, consider one of the several Direct Digital Synthesis ICs from Analog Devices. There is an excellent introduction to DDS here. 

The way DDS ICs work is: Memory within the device contains data for all supported waveforms, most commonly supported being the sine wave. The device uses an external clock or oscillator of frequency much higher than the desired frequency range, to generate the desired waveform at the desired frequency. 



Both stability and waveform purity achieved are very high, so long as recommended output filter capacitors are used. 

The Analog Devices AD9850 is a good option for the purpose described in the question, providing a highly accurate, spectrally pure analog sine wave all they way up to 40 MHz. The frequency tuning resolution is 0.0291 Hz across the entire range.

Reference schematics and even PCB layouts are provided towards the end of the datasheet.

Alternatively, if there is a requirement to test out the functioning of a DDS in a pre-built module before jumping into design, such modules are available on sites like eBay for under $5 including shipping. 



A microcontroller will be required for sending the frequency tuning commands to the DDS IC.