How would birth and death records be stored?

Originally all records would have been kept in a ledger. Families often kept their own record in a bible and churches kept their own ledgers.   The change over from private records to a state run Division for Vital Records and Health Statistics started around 1860 and only gradually became nation wide.  The US began collecting birth data at the national level in the 1902 census although it was still incomplete in 1945.

By the 1930s, when a request for a birth or death certificate was received, the information would have been taken from the ledger and hand written or typed onto a pre printed form which would have been stamped as an official copy of the contents of the ledger.  Gradually this was replaced by actual birth and death certificates held in physical file drawers similar to those that libraries use for their book records.

As an example, in Michigan the ledger entry process was replaced with the immediate filing of death certificates signed by the attending physicians and funeral director starting in 1898. The filing of birth certificates signed by the attending physician or midwife started in 1906.  Fun fact; that is how the process still works today.

In my state, birth certificates are either stored in the Office of Vital Records, Registrar of Deeds or Clerk of Courts.

In the 1930s there were no electronic forms of record keeping. Depending on where you are referring to records of births and deaths would be kept in city or country records, parish records or family bible records.

In the United States, birth and death records are stored a variety of places. Official birth records and death records are maintained at the county in which the birth or death occurred, which records can be gotten by contacting the county clerk or county courthouse. All states also maintain official birth records, requiring counties to supply them the information, and some states also maintain official death records, usually under the auspices of its health department of under the auspices of a bureau of vital statistics. Official birth and death records are also maintained by hospitals and churches, meaning their records are accepted by state governments and by the federal government as primary evidence of birth and/or death and so as official records. Certain federal agencies, like the Social Security Administration and the Census Bureau also maintain birth and death records, though not as consistently or necessarily as reliably as other sources. Birth and death records are also published and maintained by local newspapers and maintained in local libraries that keep historical copies of local newspapers, so newspapers or notarized copies of newspaper publications are used as birth and death records, too. A record of birth or death can even be gotten from what a family member wrote in a family Bible, family Bibles being acceptable proof of a person's birth or death when no other better record is available.  

Most records and all primary records of birth and death are non-electronic, including county records and state health department records. That isn't to say that an electronic copy of the record can't be found, gotten, or used, but the record itself is always a paper record, not merely electronic, if it's to be legally acceptable as primary proof of birth or death. 

With so many people born before computers even existed, making all the records electronic is a major undertaking and is always problematic in that it is merely copies data and is not the original source. Moreover, data corruption has been problematic, not just from the standpoint of hacking and electronic problems but also from the standpoint of data entry and human error. At this juncture, legal records of birth and death still require actual witnesses putting the information in writing over their actual signature on an actual piece of paper. That piece of paper may now get scanned or used for data input, but records of births and deaths are still maintained by the source generating the record, whether it be a hospital, a mortuary, a church, etc., and an original record put on a county form that they're required to furnish to the county is also maintained at the county government buildings in the county where the birth or death took place, usually with backup copies maintained in a separate site in the form of photographs of the records maintained on microfilm of microfiche, and in some places for newer records, digital pictures or scans of those actual paper records. Also, redundant copies of birth records and sometimes death records are also forwarded to the state to maintain should anything happen to the county records, like if there's a fire.