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I need data on monthly precipitation for Cape Town between 1970 and 2018. I found this page: http://cip.csag.uct.ac.za/webclient2/datasets/ But it only seems to offer information until 2000.

It's the same with this: http://worldclim.org

Then there is this one: https://gis.ncdc.noaa.gov/maps/ncei/cdo/alltimes?layers=0 But the data here is very fragmentary. Also the site is currently not working because of the US partial shutdown...

Can anyone give me a good hint of where I should look for good data?

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It really depends on what you're looking for and what data formats you want to use. If you're looking for single location direct readings from rain gauges, you might have much better luck contacting the South Africa Weather Service (though this might not be free and is typically not 'Open' in the strictest sense of the word).

Some alternatives to that really depend on what you want (gauges, vs. radar estimates vs. homogenized estimates vs. composite products), the granularity (monthly, daily, hourly; observations or normalized climatologies), the spatial resolution (point locations, grids), and the format (Spreadsheets, GRIB, CF-compliant NetCDF, etc.). It's worth also noting that radar detection of precipitation didn't really become an employed standard until the 1980's (ref.)

For a few options that are open, I would suggest looking over what's available through the NSF's National Center for Atmospheric Research . The CMAP: CPC Merged Analysis of Precipitation or the GPCP (Monthly): Global Precipitation Climatology Project might work for you.

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TrevorJ. Smith's answer sums it up nicely. SAWS data is certainly the only measured data you are likely to get and it is always preferable to use measured data over modelled data. If you request data from SAWS you will be asked to fill out and sign a disclosure statement, which will be evaluated. If they are satisfied with your request you will receive that data which provides rainfall data for every day within an available period (for a requested weather station). You can request through a SAWS Data Request Form

If your request is unsuccessful or you are happy to use modelled data I can recommend using CHIRPS data - see this link or you can view this dataset on Google Earth Engine. Unfortunately CHIRPS data is only available from 1981.

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The KNMI Climate Explorer is a great resource both for modelled data and actual station measurements. Monthly station data can be searched here.

They have data from several stations in South Africa, I pulled out this time series of precipitation for Cape Town

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You can find more info or download the date in the page for this station.

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