Uh, be careful of terminology
Watts is a point in time measurement (like speed). kWh hours is like volume (or distance, ie miles).
4000 watts a day - not a valid statement.
if that was 4kW system, that would be about half of my system size, and no where near enough to provide 32kWh of energy overnight (you'd obviously be importing most of your energy from the grid). Also beware panel ratings (lab measured) vs expected actual peak output (a lower number [think engine HP, and HP at rear wheel]) and then there is PV production ability based on panel placement (orientation towards sun, angle, etc). In other words, same panels as on my roof would produce different output on my next door neighbors house as roof is oriented differently (also curved road)
And beware Summer vs winter PV production variances. Ideally you need more kWh in summer than winter?
As for basic question... assuming using your existing freezers and HVAC, then you are talking about A/C powered devices, so DC from panels straight into battery does you no good. You have to have something to convert DC in battery to AC for your devices, an Inverter. And if you spec your inverter/system properly, yes it can charge from house outlet (grid)
Issues you need to consider/decide/answer
- where are panels to be mounted? ground mount tends to be desirable, but often not practical (depends). If roof mounting, consider lifespan of panels (often 25+ years) vs roof itself (ie don't put panels on roof that needs re-doing in near term).
- is shade on panels a consideration? some circumstances appear to be able to benefit from micro-inverters, also sometimes more cost-effective in certain smaller systems
- The energy draw of what your Inverter needs to support? ie all the loads you plan to have... have you measured the circuit usage already? identified peak draw?
As PV production drops off significantly as dusk at dawn, and *if* you want to minimize grid importing, I'm going to guess something closer to 40kWh in battery will be target. Then for your geographic location, panel orientation, panel area to work with, etc determine panel layout & quantity, with associated PV production estimates. Assuming you need to produce 32kWh per day (in peak summer, lower for rest of year), then you are targeting 32kWh of PV production on a sunny summer day. I'm guessing a 8kW PV system or larger (that peak production value is only for 1-2 hours a day).
then, if you want to avoid grid-importing during cloudy/rainy days, etc, then you need even more battery (hence my guestimate to start at 40kWh or thereabouts)... but the common adage is the cheapest energy to generate is the one you don't need in the first place (ie, sometimes (MUCH) cheaper to get more efficient lighting, HVAC, cooler units, than getting more solar PV capacity .. ymmv)
And, then you need to consider what happens with battery runs out... you ok with everything powering off? if not, then you'll be 'grid-tied' and all that implies (biggest issue tends to be cost for UL certified gear). You can go for a non-exporting 'off-grid' system, like the EG4 6000XP or 12000XP depending on inverter output requirements (ie your local load)