In homes where there are no drain lines in the basement floor, it's common practice to use some form of cistern containing a pump. A quick search at Home Depot turned up the SANISWIFT®, made by a company called Saniflo. The specifications list this unit as having a vertical discharge of 14 ft., which should be more than enough.
I've seen similar setups in many basements, where the only drain available is overhead.
Grey water from the sink and washer are collected in the cistern, then pumped up and out of the area when the cistern is filled to a certain level.
If the washer discharges faster than the grey water pump can handle, you're going to have problems. That's why when installing a system like this, the cistern should always be large enough to hold more than the volume of water that will be discharged by the washer. In the above image, the sink acts as a secondary reservoir (as well as an air gap) allowing the pump extra time to discharge the water. If you're not using a sink it's a good idea to have a larger cistern, and/or a grey water pump that can handle the volume.
You'll also want to install a check valve, on the discharge line from the grey water pipe. This will prevent the water in the vertical pipe from draining back into the cistern, and causing a potential infinite loop situation.
As for whether or not you should rely on your washing machine discharge pump to lift the water, you might find this related question useful.
WARNING:
This is only appropriate for grey water discharge. This method is NOT appropriate for sewage discharge.
Obviously there are other manufacturers of other systems, and you could always build you own system. I do not recommend or endorse this product, it is only used as an example. Searching for grey water pump will find many alternative products.
That sounds like an overly complicated setup with far more pumps than it should need. Or else you said pump when you meant tank in two places, actually, re-reading it.
In short, you can have as many pressure tanks as you like. Locating them closer to the point of use (the house) would be helpful, to a limited extent. That limited extent is the "effective volume" of the pressure tank at the house - while it's got pressure, you get better pressure at the house. As soon as it runs out of water and you become dependent on the pump, the pressure loss of the 300 feet of 1" PVC to the house comes into play, until you slow down your water use to the point where the pressure tank at the house fills up again.
If you do not change the setpoint pressure, the peak water pressure in the house will not be affected at all. Once the pressure tank at the house is depleted, the pressure will be the same as it is now. If you have a pressure gauge at the house now (or can add one) it might be useful to know the pressure at the house when "loads of interest" (showers, say) are drawing water.
Most pump controls can be altered to have a higher setpoint pressure, and most well pumps will provide that, within reason. General recommendations are to stay below 80 PSI - but if you are now running your pump on at 20 off at 40 (a typical default) you could turn it up to on at 30 off at 50, or on at 40 off at 60 - you will need to adjust your pressure tank (and any you may choose to add) for the higher pressure, particularly if they are bladder-type tanks - with the system drained, a bladder tank should typically be 2 PSI less than the low water pressure setpoint (ie, 18, 28, or 38 for the three ranges I've just given) and you may need additional pressure tank volume (because the effective volume of water a pressure tank can hold goes down as the system pressure goes up.) So, you can probably get more pressure without another pump, and possibly without another tank, but we'd need more details of what your system is doing now to know that for certain.
Unless your water use is extreme, 300 feet of 1" PVC pipe should not have a lot of effect on the pressure - at 5 gallons per minute, about 2.2 PSI - at 10GPM, 8.2PSI, 15 GPM, 17.4 psi
Best Answer
Two factors matter: flow rate and pressure. Pressure is determined by friction loss and elevation adjustment. Since the tank and house are at the same elevation, we don't need to account for that. We do need to account for your friction loss, though. To do that, we need to know what your pipe diameter, length, and flow rate.
We start with flow rate because the amount of water flowing through a pipe determines its friction loss. That's what we size the pipe from. Your flow rate should be based on maximum flow: everything that could possibly ever run at one time. Your shower, dishwasher, washing machine, somebody washing their hands, etc. There's no uniform minimum standard for that, so check the fixtures around your building to find out how muh they would draw.
According to a plumbing design guide from Michigan, this can vary from 7 gpm for one bathroom to 17 gpm for a 3-4 bathroom residence. After you've determined the flow rate, you must select a desired pressure for the house inlet pressure.
If your pump is able to fully supply your house without the aid of a tank (sufficient flow at operating pressure), then your only consideration for tank size is the cycle time of the pump. For smaller pumps, you want your tank to provide at least 2 minutes of water. For larger pumps, you may desire a higher cycle time. There is no harm in a larger tank size.
If your pump isn't able to provide that level service, then you'll need to size your tank to provide supply for a period of time. Remember that pump supply in gpm is also related to pressure. The more pressure the pump needs to provide, the less flow it will have. Calculate your pump flow based on the pressure needed to charge the tanks.
The ultimate calculation is (maximum flow * maximum minutes) > (tank capacity / minutes + pump capacity * minutes). You can use any combination of tank and pump capacity as long as the tank + pump capacity meets your demand needs. A 1 gpm pump and a pressure tank that can supply 300 gallons could probably hold you over just as well as a 14 gpm pump and 28 gallons of supply tank. Remember again that supply is not equal to full capacity: you only have supply if it is higher than your necessary pressure. Your goal is to find a combination that is most cost effective.
For another good resource related to tanks, see Flotec's pre-charged pressure tank FAQs.
By the way, since you gain 1 psi for every 2.31 feet of elevation gain (referred to for some reason as "feet of head"), you can get a lot of "free" tank supply pressure by putting your tank at a higher elevation, though that will increase your pump charging pressure.
For absolute luxury, keep your entire supply system able to supply above your desired household pressure and slap a pressure reducing valve on the supply line just before it reaches the first appliances. 60 psi is the ideal dream coming out of your pressure valve (fantastic showers!), but pressure does cost in terms of electric usage for lower pump capacity because, again, higher pressure means lower flow. Most systems are designed to start charging at 40, 30, or 20 psi, though. Note how the pressure corresponds to the amount of water that can be supplied in this example tank.