The storage of data in DNA typically involves encoding and synthesizing data into short oligonucleotides, followed by reading with a sequencing instrument. Major challenges include the molecular consumption of synthesized DNA, basecalling errors, and limitations with scaling up read operations for individual data elements. Addressing these challenges, we describe a DNA storage system called MDRAM (Magnetic DNA-based Random Access Memory) that enables repetitive and efficient readouts of targeted files with nanopore-based sequencing. By conjugating synthesized DNA to magnetic agarose beads, we enabled repeated data readouts while preserving the original DNA analyte and maintaining data readout quality. MDRAM utilizes an efficient convolutional coding scheme that leverages soft information in raw nanopore sequencing signals to achieve information reading costs comparable to Illumina sequencing despite higher error rates. Finally, we demonstrate a proof-of-concept DNA-based proto-filesystem that enables an exponentially-scalable data address space using only small numbers of targeting primers for assembly and readout.