Is the ESP8266 able to collect 8 bytes data at 1 KHz for 1 second through I2C?
The limiting factor here is the i2c transmission rate, not the CPU of the ESP8266.
As Kevin points out, there is overhead in i2c transmission. The i2c master (CPU) has to set a register, then send a read request, before clocking out the 6 bytes of acceleration data (3 axes x 16 bits per axis) from the i2c slave (sensor). So your estimate of 8kB/sec is about right.
Standard i2c at 100kHz can handle about 10kB/sec. That's cutting it too close. If the sensor-to-CPU PCB leads are short, you can use 400kHz i2c. It's just a configuration setting in the CPU firmware. The sensor has a 1kB FIFO buffer that will help if the CPU gets busy with some other task and can't maintain a 1000/sample/sec reading rate.
The ESP8266 should have enough computing power to request and store the sensor data to RAM, but it may get interrupted by other tasks for the radio, clocks, calculations, etc. You'll have to carefully write the software to prioritize collection of the sensor data stream, and put off any noncritical tasks while the data is coming in. Good luck!