If you drive the heaters with a voltage source which has no current limiting when the tube powers up the heaters are cold and there is a surge in current which reduces the life of the heaters. A 3-terminal regulator limits the current because the regulator has a current limit.
The LM317 voltage regulator EBK posted would work since it has current limiting. Power dissipation = (9-6.3)*0.3 = 0.81W. The TO-220 package has a thermal resistance of say 50 degC/W so the temperature rise will be 0.81 * 50 = 41 degC. So it will run hot but you will probably get away with it and it would be wise to locate the regulator away from the tube itself.
The simplest way to power the heaters with with a series resistor. 9V in and 6.3V out at 300mA means the resistor needs to be R = (9-6.3) / 0.3 = 9 ohms. The resistor will dissipate 0.3^2 * 9 = 0.81W. The 9 ohm resistor limits then power-on surge current as well.
An issue is if the socket is dirty an one of the heater contacts doesn't connect then the 300mA goes down one of the heaters. The way around that is easily fixed. Use an 18 ohm resistor for each heater. (9 - 6.3) / 18 = 150mA per tube.
Also, if you accidentally plug in 12V then the heater current will be more than 300mA. The voltage regulators stop that.
Another way is to power the heaters via a current source. There's no power-up surge at all and it doesn't care what the input voltage is.
There's plenty of posts on the web about these.
Unfortunately a simple current using a LM317 regulator doesn't quite work on 9V when you consider the regulator drop out. It might work in practice but on paper there's not enough input voltage for the current to regulate properly. (There's no need for a heatsink because the regulator and the current sense resistors share the power dissipation.)
[No good]

These types are also simple but you have to be careful of temperature dependency,

[The single current source doesn't solve the dirty socket problem.]
So least parts and headaches is probably EBK's LM317.