Airbus started with the A300, which as usernumber commented was a 300-seat airliner design. So A for Airbus, and 300 for the number of seats. An A250 was also considered (guess how many seats), which became the A300B. The models counted up by 10 from there.
Boeing model numbers historically counted up, including the Model 40, Model 80, Model 247, Model 307 Stratoliner and Model 377 Stratocruiser. At the dawn of the jet age, they decided to change up their numbering scheme:
To support this diversification strategy, the engineering department divided the model numbers into blocks of 100 for each of the new product areas: 300s and 400s continued to represent aircraft, 500s would be used on turbine engines, 600s for rockets and missiles and 700s were set aside for jet transport aircraft.
The 707 prototype was called the 367-80, or "Dash 80." When the time came to give it an official number they knew it was going to be a "Jet Transport" and would get a 700-series number, and they decided "707" had a better sound to it than "700". They kept the 7's and have counted up from there (not entirely in order).